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Radical Equation Calculator

Enter your radical equation via text, photo, or camera, and get step-by-step solutions for square, cube, and nth roots.

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Radical Equation Calculator

Our Radical Equation Calculator allows you to solve equations involving radicals step by step. Enter your equation to see detailed solutions, learn how to simplify radicals, and eliminate them effectively.

What Is a Radical Equation?

Definition: A radical equation is an equation in which the variable appears inside a root, such as a square root, cube root, or higher-order root.

Example: √(x + 9) = 6

The word "radical" comes from the Latin radix, meaning "root." In math, radicals indicate roots of numbers or algebraic expressions.

Standard Form of a Radical Expression:

n√a

Where:

√ = radical sign
a = radicand (value inside the radical)
n = index (degree of the root)

Steps to Solve a Radical Equation

Isolate the radical on one side of the equation.
Raise both sides to the power of the radical's index to remove the radical.
Simplify and solve the resulting equation for the variable.
Always check your solutions in the original equation to eliminate extraneous roots.

Example Problem:

Solve: √(x + 7) = x - 3

Step 1: Radical Isolated

√(x + 7) = x - 3

Step 2: Square Both Sides

(√(x + 7))² = (x - 3)² → x + 7 = x² - 6x + 9

Step 3: Rearrange into Quadratic Form

x² - 7x + 2 = 0

Step 4: Solve Quadratic

x = (7 ± √(49 - 8))/2 = (7 ± √41)/2

Step 5: Verify Solutions

Check each root in the original equation to discard extraneous solutions.

Using the Radical Equation Calculator

Enter the Equation: Type or paste your radical equation. You can also upload an image of the equation.
Click Solve: The calculator processes the equation.
View Results: Step-by-step solutions with verification of extraneous roots.
Extra Features: Reset clears the input; Load Example fills the box with sample problems.

Properties of Radicals

1. Product Rule

√(a) × √(b) = √(a × b)

Example: √2 × √8 = √16 = 4

2. Quotient Rule

√(a) / √(b) = √(a / b), b ≠ 0

Example: √49 / √7 = √7 = √7

3. Power Rule

n√(a^m) = a^(m/n)

Example: ³√(27^2) = 27^(2/3) = 9

4. Radical of a Radical

m√(n√a) = (mn)√a

Example: √(√81) = ⁴√81 = 3

5. Combining Like Radicals

a√b + c√b = (a + c)√b

Example: 2√5 + 3√5 = 5√5

6. Multiplying Radicals

√a × √a = a

Example: √6 × √6 = 6

7. Simplifying Radicals

√(a^2 × b) = a√b

Example: √(50x^2) = 5x√2

FAQs

1. Radical Expression vs Radical Equation?

A radical expression lacks an equal sign, while a radical equation has one.

2. When Does a Radical Equation Have No Solution?

No solution occurs if the radicand is negative for an even root or if the solution is extraneous.

3. Does the Root Index Affect Validity?

Even index: Radicand ≥ 0

Odd index: Radicand can be any real number

Example: ³√(-8) = -2

References

LibreTexts: Radical Equations
Khan Academy: Radical Equations & Functions
Lumen Learning: Solving Radical Equations

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